Modulated wave trains in generalized Kuramoto-Sivashinksi equations

TL;DR

This paper analyzes the stability of periodic wave trains in a generalized Kuramoto-Sivashinski equation, linking spectral stability to Whitham's modulation equations and providing insights into low frequency perturbations in thin film flows.

Contribution

It derives and utilizes Whitham's modulation equations to connect spectral stability with modulation properties in the gKS equation.

Findings

Spectral stability at small wavenumber is always critical due to translational invariance.

The modulation equations determine the behavior of low frequency perturbations.

The study relates spectral stability to properties of the modulation equations.

Abstract

This paper is concerned with the stability of periodic wave trains in a generalized Kuramoto-Sivashinski (gKS) equation. This equation is useful to describe the weak instability of low frequency perturbations for thin film flows down an inclined ramp. We provide a set of equations, namely Whitham's modulation equations, that determines the behaviour of low frequency perturbations of periodic wave trains. As a byproduct, we relate the spectral stability in the small wavenumber regime to properties of the modulation equations. This stability is always critical since 0 is a 0-Floquet number eigenvalue associated to translational invariance.